We introduce the following auxiliary construction, which is interesting in its own right.
Let \({\operatorname{Ex}}\colon {{S}\mathrm {et}_{\hspace{-0.85355pt}\Delta }}\rightarrow {{S}\mathrm {et}_{\hspace{-0.85355pt}\Delta }}\)  denote the right adjoint to the simplicial subdivision functor \(\operatorname{Sd}\) —recall that \(\operatorname{Sd}\)  sends \(\Delta ^n\)  to the nerve of the category of totally ordered subsets of \([n]\) .
For instance, \(\operatorname{Sd}\Delta ^1\)  can be sketched as the cospan \(\lbrace 0\rbrace  \rightarrow \lbrace 0,1\rbrace  \leftarrow \lbrace 1\rbrace \) .
Our main reference on \({\operatorname{Ex}}\)  is [1]}.
The functor \({\operatorname{Ex}}\)  comes with a natural weak equivalence \(b \colon 1_{{S}\mathrm {et}_{\hspace{-0.85355pt}\Delta }}\overset{\sim }{\longrightarrow }{\operatorname{Ex}}\)  (see, for instance, [1]}).
